The discussion revolves around evaluating the limit of the expression (a^x - a^-x - 2)/(x^2) as x approaches 0. Participants are exploring connections to known limits, particularly the special limit involving e^x.
Participants are actively questioning the setup of the problem, with one suggesting a potential typo in the original limit statement. There is a mix of approaches being considered, including algebraic manipulation and the use of series expansions, but no consensus has been reached on a definitive method.
One participant notes that L'Hospital's rule may not apply due to the numerator not equating to zero at x = 0, leading to a discussion about the implications of this observation on the limit's value.
alingy1 said:Homework Statement
lim of (a^x-a^-x-2)/(x^2) as x->0
Find the value of the limit^.
The answer is (lna)^2
I know how to get the answer using L'Hospital.
However, I do not want to use Hospital's rule.
I think I see the pattern with the special limit:
lim of (e^x-1)/x as x->0
But I just don't see how I can apply that. I tried to take ln() out of everything, but it leads me to nowhere.
Please help me.
vela said:I think there's a typo in the original post. The problem should be
$$\lim_{x \to 0} \frac{a^x + a^{-x} - 2}{x^2}.$$ You can rewrite this as
$$\lim_{x \to 0} \frac{a^{-x}[(a^x)^2 - 2a^x +1]}{x^2}.$$ You can do a little algebra, use what Dick said about expressing this in terms of ##e##, and change variables appropriately to relate it back to the special limit.